Asymptotic results for random polynomials on the unit circle

  1. Gabriel H. Tucci
  2. Philip Whiting

Abstract

In this paper we study the asymptotic behavior of the maximum magnitude of a complex random polynomial with i.i.d. uniformly distributed random roots on the unit circle. More specifically, let {nk}k=1 be an infinite sequence of positive integers and  let {zk}k=1 be a sequence of i.i.d. uniformly distributed random variables on the unit circle. The above pair of sequences determine a sequence of random polynomials PN(z)Nk=1 (z − zk)nk with random roots on the unit circle and their corresponding multiplicities. In this work, we show that subject to a certain regularity condition on the sequence {nk}k=1, the log maximum magnitude of these polynomials scales as sNI, where s2NNk=1 nk2 and I is a strictly positive random variable.

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Probability and Mathematical Statistics

34, z. 2, 2014

Pages from 181 to 197

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